Gaseous system: Meaning of this integral eq.

In summary: However, this may not be the same as the peak frequency \omega_{0}. The point \vartheta = a can also tell us the standard deviation of the distribution, which provides information about the spread or variability of the data. Overall, the integral has a statistical meaning in that it is used to calculate the probability of a certain value occurring in a Gaussian distribution.
  • #1
jam_27
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0
What is the physical or statistical meaning of the following integral

[tex]\int^{a}_{o} g(\vartheta) d(\vartheta)[/tex] = [tex]\int^{\infty}_{a} g(\vartheta) d(\vartheta)[/tex]

where [tex]g(\vartheta)[/tex] is a Gaussian in [tex]\vartheta[/tex] describing the transition frequency fluctuation in a gaseous system (assume two-level and inhomogeneous) .

[tex]\vartheta = \omega_{0} -\omega[/tex], where [tex]\omega_{0}[/tex] is the peak frequency and [tex]\omega[/tex] the running frequency.

I understand that the integral finds a point [tex]\vartheta = a[/tex] for which the area under the curve (the Gaussian) between 0 to a and a to [tex]\infty[/tex] are equal.

But is there a statistical meaning to this integral? Does it find something like the most-probable value [tex]\vartheta = a[/tex]? But the most probable value should be [tex]\vartheta = 0[/tex] in my understanding! So what does the point [tex]\vartheta = a[/tex] tell us?

I will be grateful if somebody can explain this and/or direct me to a reference.

Cheers

Jamy
 
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  • #2
angThe integral in question is a type of probability distribution known as a Gaussian or normal distribution. This type of distribution is used to describe the probability of a certain event occurring. In this case, the integral is computing the probability of the transition frequency fluctuation in a gaseous system taking on a particular value (in this case, \vartheta = a). The integral finds a point \vartheta = a for which the area under the curve (the Gaussian) between 0 to a and a to \infty are equal. This is the most probable value of \vartheta - i.e. the value that is most likely to be observed.
 
  • #3


The integral in this equation has a physical meaning in the context of a gaseous system. It represents the probability of finding a transition frequency fluctuation, described by a Gaussian function g(\vartheta), within a certain range of values. In this case, the range is from 0 to a and from a to \infty. This integral can also be thought of as the cumulative distribution function (CDF) of the Gaussian function, which gives the probability of a random variable falling within a certain range of values.

The point \vartheta = a represents the boundary between the two regions of the CDF, where the probabilities of finding a transition frequency fluctuation are equal. This means that there is a 50% chance of finding a fluctuation within the range of 0 to a and a 50% chance of finding it within the range of a to \infty.

In terms of statistical meaning, this integral can be used to calculate the standard deviation of the transition frequency fluctuations in the gaseous system. The value of a represents the standard deviation of the Gaussian distribution, which is a measure of the spread of the data points around the mean value (in this case, \vartheta = 0).

Overall, this integral provides insight into the distribution of transition frequency fluctuations in a gaseous system and can be used to calculate important statistical parameters. I would suggest consulting a textbook or scientific paper on statistical mechanics for further information and references.
 

FAQ: Gaseous system: Meaning of this integral eq.

What is a gaseous system?

A gaseous system refers to a collection of gas molecules that are in a specific container or space. The behavior and properties of the gas molecules in the system can be described using different mathematical equations.

What is the meaning of an integral equation in relation to a gaseous system?

An integral equation is a mathematical representation that describes the relationship between different variables in a gaseous system. It helps to understand the behavior of the gas molecules and their interactions with each other and the container.

How is an integral equation used to study gaseous systems?

An integral equation is used to solve for unknown variables and understand the behavior of gas molecules in a system. It can also be used to predict changes in the system under different conditions and to compare different gas systems.

What are the key components of an integral equation for a gaseous system?

The key components of an integral equation for a gaseous system include the pressure, volume, temperature, and number of moles of gas molecules present in the system. These variables are used to describe the behavior of the gas molecules and their interactions.

Are there different types of integral equations for gaseous systems?

Yes, there are different types of integral equations used to describe gaseous systems, such as the ideal gas law, van der Waals equation, and the Boltzmann distribution. Each equation has its own set of assumptions and is used in different scenarios to study gaseous systems.

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